A215914 Smallest positive integer k such that there is no k-dimensional unital *-subalgebra of the n X n complex matrices.

2, 3, 4, 7, 8, 13, 16, 23, 24, 43, 44, 49, 64, 77, 80, 97, 116, 141, 144, 167, 168, 193, 248, 249, 280, 313, 348, 385, 424, 473, 484, 527, 528, 573, 620, 625, 720, 725, 828, 833, 890, 949, 1010, 1073, 1088, 1153, 1220, 1289, 1360, 1433

Offset: 1

Author

Nathaniel Johnston, Aug 26 2012

Keywords

• nonn

Comments

a(n) is the smallest positive integer that is not contained in the n-th row of A215905.

a(n) >= n+1. In fact, for any m >= 1 there exists N >= 1 such that a(n) > mn for all n >= N. That is, this sequence grows super-linearly.

Examples

			In the n = 4 case, there are unital *-subalgebras of dimensions 1 though 6, as follows:
a 0 0 0 ... a 0 0 0 ... a 0 0 0 ... a 0 0 0 ... a 0 0 0 ... a 0 0 0
0 a 0 0 ... 0 b 0 0 ... 0 b 0 0 ... 0 b 0 0 ... 0 a 0 0 ... 0 b 0 0
0 0 a 0 ... 0 0 b 0 ... 0 0 c 0 ... 0 0 c 0 ... 0 0 b c ... 0 0 c d
0 0 0 a ... 0 0 0 b ... 0 0 0 c ... 0 0 0 d ... 0 0 d e ... 0 0 e f
However, there is no unital *-subalgebra of the 4-by-4 matrices of dimension 7, so a(4) = 7.
		

Crossrefs

Cf. A215905, A215909.